Calibrations for minimal networks in a covering space setting

In this paper we define a notion of calibration for an equivalent approach
to the classical Steiner problem in a covering space setting and we give
some explicit examples.
Moreover we introduce the notion of calibration in families:
the idea is to divide the set of competitors in a suitable way,
defining an appropriate (and weaker) notion of calibration.
Then, calibrating the candidate minimizers
in each family and comparing their perimeter, it is possible to find the minimizers of the minimization problem.
Thanks to this procedure we prove the minimality of the Steiner configurations spanning
the vertices of a regular hexagon
and of a regular pentagon.